Series 2: Environmental Systems modelling

Series Overview

Environmental systems operate through conservation laws—mass, energy, momentum. This series develops process-based models from first principles: radiation balance, heat transfer, water movement, biogeochemical cycling. Students learn to construct predictive models of environmental dynamics rather than merely describing patterns.

Pedagogical Philosophy

Conservation laws as organizing principles. Every environmental process conserves something. We derive models by writing balance equations: inputs minus outputs equals change in storage. This provides conceptual unity across disparate phenomena.

Dimensional analysis as verification. Units must balance. Every term in every equation checked for dimensional consistency. This builds physical intuition and catches errors.

Process models predict, not just describe. Unlike statistical models, these equations encode mechanisms. Parameters have physical meaning. Models generalize beyond calibration conditions.

Learning Objectives

Derive energy balance equations for surfaces, canopies, water bodies
Calculate radiative, sensible, and latent heat fluxes
Model water movement through infiltration, runoff, groundwater flow
Quantify evapotranspiration via energy- and resistance-based methods
Simulate biogeochemical cycles (carbon, nitrogen, phosphorus)
Apply conservation principles to novel environmental systems
Parameterize models from field measurements
Validate predictions against observations

Model Sequence

Cluster F: Energy Balance (Models 13-16)

Model 13: Net Radiation and Surface Energy Balance Shortwave/longwave radiation. Albedo. Emissivity. Stefan-Boltzmann law. Surface temperature prediction.

Model 14: Soil Heat Flux and Thermal Properties Heat conduction. Thermal diffusivity. Diurnal temperature waves. Depth of penetration.

Model 15: Sensible Heat Transfer Temperature gradients. Aerodynamic resistance. Bulk transfer coefficients. Bowen ratio.

Model 16: Latent Heat and Evapotranspiration Phase change energy. Penman equation. Penman-Monteith model. Stomatal resistance.

Cluster G: Hydrological Processes (Models 17-22)

Model 17: Infiltration and Green-Ampt Model Soil moisture movement. Capillary action. Wetting front propagation. Time to saturation.

Model 18: Runoff Generation Mechanisms Infiltration-excess vs saturation-excess. Variable source area. Hydrograph components.

Model 19: Groundwater Flow (Darcy’s Law) Hydraulic conductivity. Gradient. Aquifer properties. Well drawdown equations.

Model 20: Streamflow Routing Continuity equation. Manning equation. Kinematic wave approximation. Flood wave propagation.

Model 21: Snowmelt Processes Energy balance for snow. Degree-day models. Rain-on-snow events.

Model 22: Evapotranspiration Partitioning Soil evaporation vs transpiration. Crop coefficients. Root water uptake.

Cluster H: Ecosystem Processes (Models 23-28)

Model 23: Photosynthesis and Primary Production Light response curves. CO₂ fixation. Gross vs net primary production. Carbon balance.

Model 24: Respiration and Decomposition Temperature dependence (Q₁₀). Michaelis-Menten kinetics. Litter decomposition rates.

Model 25: Nitrogen Cycling Mineralization. Nitrification. Denitrification. N₂O emissions. Mass balance models.

Model 26: Carbon Cycle Dynamics Pools and fluxes. Residence times. Steady-state vs transient behavior.

Model 27: Leaf Area Index and Canopy Structure Beer’s law in canopies. Light interception. LAI measurement methods.

Model 28: Ecosystem Water Use Efficiency Carbon gain per water lost. Intrinsic vs integrated WUE. Climate change implications.

Mathematical Progression

Energy balance: Algebraic equations (fluxes), first-order ODEs (temperature change)

Hydrology: PDEs (diffusion equation), coupled ODEs (reservoir routing)

Ecosystems: Coupled nonlinear ODEs (biogeochemistry), optimization (resource allocation)

Computational Skills

ODE solvers (explicit Euler, RK4)
PDE discretization (finite differences)
Mass balance accounting
Parameter optimization (least squares)
Sensitivity analysis
Monte Carlo uncertainty propagation

Prerequisites

Required: Series 1 (differential equations, exponential functions)

Helpful: Basic physics (energy concepts), chemistry (moles, reactions)

Entry Points by Background

Environmental science: Start Model 13. Core material for major.

Hydrology focus: Models 17-22 self-contained after Series 1.

Ecology students: Models 23-28 after completing 13-16 (energy background).

Engineering (environmental): Familiar physics; focus on spatial/ecosystem applications.

Key Insights

Energy drives hydrological processes. Evaporation, snowmelt, soil thawing all require energy budget.
Water couples energy and carbon cycles. Transpiration links photosynthesis to energy balance.
Nonlinearity everywhere. Exponential temperature dependence. Threshold behaviors. Positive feedbacks.
Parameters vary in space and time. Soil properties, vegetation characteristics, weather forcing all heterogeneous.
Scale matters. Plot-scale vs watershed-scale processes differ fundamentally.
Closure assumptions necessary. More unknowns than equations; must parameterize subgrid processes.

Applications Covered

Irrigation scheduling (ET estimation)
Flood forecasting (runoff generation + routing)
Carbon accounting (NEE calculation)
Drought prediction (soil moisture modelling)
Snowmelt runoff timing
Agricultural emissions (N₂O from fertilizer)
Lake/reservoir thermal structure

Extensions

For climate: Energy balance models foundational for climate dynamics (future series).

For cryosphere: Series 4 extends energy/water concepts to frozen systems.

For remote sensing validation: Series 5 uses these process models for ground truth.

Estimated Time

Per model: 3-4 hours

Full series: 50-65 hours

Core energy-water sequence (13-16, 17-18, 20, 22): 25-35 hours

Prerequisites: Complete Series 1. Models 13-16 provide foundation for rest of series.